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Video: The Probability of the Union of Mutually Exclusive Events and the Complement Rule

Tim Burnham

Learn how to calculate the probability of one or other mutually exclusive events occurring and use the idea that either an event or its complement is certain to occur.

08:48

Video Transcript

When events are mutually exclusive, they can’t happen together. If one happens, then the other can’t. For example, if I flip a coin, it can’t land head side up and tail side up at the same time, only one or the other. In this video, we’re gonna calculate the probabilities of unions of mutually exclusive events.

Let’s imagine that we’ve got a bag with ten discs in it numbered one to ten. Some of these discs are blue, some are orange, and some are green. If I shake up the bag and I pick a disc at random, it could be blue or green or orange. It must be just one of these colors.

So the events pick a blue disc, pick an orange disc, and pick a green disc are mutually exclusive. If one of them happens, none of the others can happen. Similarly, the number on the disc which we randomly pick could be odd or it could be even, but not both. The events pick an odd number and pick an even number are also mutually exclusive.

Now let’s think of some different events. What about combinations like pick blue and pick even? So events like pick a blue disc and pick an even number could both happen together because there is a blue disc which is an even number. So for example, disc two is blue and it’s even, so blue and even are not mutually exclusive events.

So one of the main objectives of this video is this rule here of mutually exclusive events. So if we know that some events are mutually exclusive, then if we want to calculate the probability of one or other of them occurring, we just have to add their probabilities together. The probability of the union of mutually exclusive events can be calculated by summing their individual probabilities. So let’s say 𝐴 and 𝐵 are mutually excusiv- exclusive events. Then the probability of 𝐴 union 𝐵, that is, the probability that 𝐴 happens or 𝐵 happens, is simply the probability of 𝐴 plus the probability of 𝐵.

Another example of mutually exclusive events are complementary events. This is when we consider the case where something either happens or it doesn’t happen.

So here’s a question. Let 𝐸 be the event I get a multiple of three when I roll a fair six-sided dice. When you roll a dice, the number that comes up is either a th- multiple of three or it isn’t a multiple of three; it can’t be both. So those two things are mutually exclusive events, but they’re a bit of a special case because the events are purely defined in terms of being something or not being that thing. And to no- to denote this not being that something, we use a dash after the letter representing the event being that thing. This is sometimes called prime or the complement of, so let’s have a look at that.

So we can write then 𝐸 dashed or 𝐸 prime or 𝐸 complement, however you want to say it, is the event I don’t get a multiple of three when I roll a fair six-sided dice. Now it’s certain that either event 𝐸 or 𝐸 dashed will occur; either I will get a multiple of three when I roll this dice or I won’t get a multiple of three, that much we know. So if I add those two probabilities together, I must get one. So the probability of 𝐸 plus the probability of 𝐸 dashed is equal to one.

And by rearranging that a little bit, if I know the probability of 𝐸, I can work out the probability of 𝐸 dash. If I know the probability of 𝐸 dash, then I can work out the probability of 𝐸. And this result becomes very useful in lots of different cases, so let’s look at an example.

Let’s say we roll two fair six-sided dice and add their scores together. We can record a sample space in a table that looks like this. So let’s define 𝐸 as being the event that the result sums to ten. And to work out the probability of event 𝐸, we just need to work out how many of the thirty-six equally likely outcomes result in a sum of ten, so we’ve got one two three.

So the probability of event 𝐸 occurring is three out of thirty-six, so that does rely on the fact that we knew that they were all equally likely outcomes because both of those are fair six-sided dice, so all of those thirty-six outcomes are equally likely.

Now because we defined 𝐸 to be the event that the result sums to ten, 𝐸 dash must be the event that the result doesn’t sum to ten. And if we want to work out that probability, we’ve got two ways of going. One, we could count up all of the cases where the result isn’t ten, so that means going through and circling all of those.

And there were thirty-three of those, so we can say the probability of 𝐸 dash is thirty-three over thirty-six. So that’s a perfectly acceptable method, but I did pause the video in order to circle all of those and actually that took quite a long time, but there is another way that we could have gone about doing this.

So remember the probability of 𝐸 dash is equal to one minus the probability of 𝐸. So that’s one minus our first answer, one minus three over thirty-six, which gives us the same answer we’ve just got, thirty-three over thirty-six, so two different ways. But the complement method is sometimes just an easier way of doing these things; it’s a quicker way than adding up all the cases where something isn’t true. So let’s look at one more example of that, which makes the point a little bit more clearly.

Okay so if we flipped five coins and wanted to know the probability that at least one of them lands tail side up, then that’s actually quite tricky. There are loads of combinations of the first, second, third, fourth, and fifth coin with any one or two or three or four or all five of them landing tail side up, so that’s quite difficult to plot out and to keep track of and to set up the probability model for that. However, if we say that either at least one coin lands tail side up or it doesn’t, then we can quickly see that there’s only one way of no coins landing tail side up, and that’s if they all land head side up, and that’s much easier to work at that probability. So let’s have a look at that.

So it’s absolutely certain, that’s a probability of one, that either we’ll get a tail or two or three or four or five tails, so we’ll get at least one tail, or we’ll get no tails at all. So the sum of those two probabilities is equal to one.

So just subtracting the probability of no tails from both sides, I can see the probability of getting at least one tail is one minus the probability of no tails. And the probability of no tails is the same as the probability of all five coins landing heads side up.

So that probability is half times a half times a half times a half times a half; it’s gonna be heads then heads then heads then heads then heads and so on, which is one over thirty-two, so the probability we get at least one tail is one minus one over thirty-two, which is thirty-one over thirty-two. So this method of complementary probabilities, the complementary rule or the complement rule, made this question much much easier to do than trying to work out all those different combinations of one tail on the first, one tail on the second, one tail on the third, one tail on the fourth, two tails in any combination, three tails in any combination, and so on.

So just a quick summary of what we’ve learned there, if 𝐴 and 𝐵 are mutually exclusive events, then the probability of 𝐴 union 𝐵 is the probability of 𝐴 plus the probability of 𝐵. So if they’re mutually exclusive events and we want to know if either one or the other happens, we just have to add their probabilities together.

And if 𝐸 is an event, then 𝐸 dashed or 𝐸 prime or 𝐸 complement is the complementary event in which 𝐸 does not occur. And we know that it’s certain that either 𝐸 will happen or it won’t happen, so the sum of those two probabilities must be equal to one. Probability of 𝐸 plus the probability of 𝐸 dash is equal to one. We can rearrange that in different ways to work out the probability of 𝐸 is one minus the probability of 𝐸 dashed, and the probability of 𝐸 dashed is one minus the probability of 𝐸. So the probability that something happens is one minus the probability that it doesn’t happen, or the probability that it doesn’t happen is one minus the probability that it does happen.